176edo
Template:Infobox ET The 176 equal divisions of the octave (176edo), or the 176(-tone) equal temperament (176tet, 176et) when viewed from a regular temperament perspective, is the equal division of the octave into 176 parts of about 6.82 cents each, a size close to 243/242, the rastma.
Theory[edit]
176edo is consistent to the 11-odd-limit, tempering out 78732/78125 (sensipent comma) and | 41 -20 -4 > (undim comma) in the 5-limit; 6144/6125, 10976/10935, and 50421/50000 in the 7-limit; 441/440, 3388/3375, 6912/6875, 8019/8000, 9801/9800 and 16384/16335 in the 11-limit. Using the patent val, 351/350, 364/363, 2080/2079, 2197/2187, and 4096/4095 in the 13-limit.
It supports the bison temperament and the commatic temperament, and provides the optimal patent val for countermiracle in the 7- and 11-limit, and countermanna, one of the extensions, in the 13-limit.
Prime harmonics[edit]
Regular temperament properties[edit]
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | | 279 -176 > | [[[:Template:Val]]] | -0.100 | 0.100 | 1.47 |
| 2.3.5 | 78732/78125, | 41 -20 -4 > | [[[:Template:Val]]] | -0.400 | 0.432 | 6.34 |
| 2.3.5.7 | 6144/6125, 10976/10935, 50421/50000 | [[[:Template:Val]]] | -0.243 | 0.463 | 6.79 |
| 2.3.5.7.11 | 441/440, 3388/3375, 6144/6125, 8019/8000 | [[[:Template:Val]]] | -0.250 | 0.414 | 6.08 |
| 2.3.5.7.11.13 | 351/350, 364/363, 441/440, 2197/2187, 3146/3125 | [[[:Template:Val]]] | -0.123 | 0.473 | 6.93 |
Rank-2 temperaments[edit]
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 17\176 | 115.91 | 77/72 | Mercy / countermiracle / counterbenediction / countermanna |
| 1 | 35\176 | 238.64 | 147/128 | Tokko |
| 1 | 65\176 | 443.18 | 162/125 | Sensipent |
| 1 | 73\176 | 497.73 | 4/3 | Gary / cotoneum |
| 1 | 83\176 | 565.91 | 13/9 | Tricot / trident |
| 2 | 23\176 | 20.45 | 81/80 | Commatic |
| 2 | 23\176 | 156.82 | 35/32 | Bison |
| 4 | 73\176 (15\176) |
497.73 (102.27) |
4/3 (35/33) |
Undim |
| 8 | 73\176 (7\176) |
497.73 (47.73) |
4/3 (36/35) |
Twilight |
| 8 | 83\176 (5\176) |
565.91 (34.09) |
168/121 (55/54) |
Octowerck (176f) / octowerckis (176) |
| 11 | 73\176 (7\176) |
497.73 (47.73) |
4/3 (36/35) |
Hendecatonic |
| 22 | 73\176 (1\176) |
497.73 (6.82) |
4/3 (385/384) |
Icosidillic |